The tangent to the curve y=x^3-2x^2+4 at the point (2,4)

Question

meets the coordinate axes at the points X and Y.What is he area of the triangle OXY?

Answer 1

The curve y = x³ - 2x² + 4

Differentiating  on each side with respect of x.

y' = 3x² - 4x

Substitute the value x = 2 in above equation.

y' = 3(2)² - 4(2)

y' = 12 - 8

y' = 8

This is the slope of tangent line to the curve at (2, 4).

To find the tangent line equation, substitute the values of m = 4 and (x, y ) = (2, 4).  in the slope intercept form of an equation y = mx + b.

4 = 4(2) + b

b = 4 - 8

b = - 4

Substitute m = 4 and b = - 4 in y = mx + b.

Tangent line is y = 4x - 4

Find the points where the tangent line meets coordinate axis.

Find the intercepts.

Substitute x = 0 in y = 4x - 4

y = 4(0) - 4

y = -4

y intercept is  - 4.

Substitute y = 0 in y = 4x - 4

4x - 4 = 0

4x = 4

x = 1

The tangent line meets the x axis at 1 and y axis at - 4.

oxy is a right angle triangle.

Area of right angle triangle = ab/2

a, b are legs of triangle.

a = 1unit , b = 4units

Area of triangle = ab/2

= 1/2(1)(4)

= 4/2

= 2

Area of triangle is 2 square units.